Solution b the depth of a circuit to rerepresent

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Unformatted text preview: Solution: Two butterﬂies back to back. 6.896 Final Examination Name 5 Problem 3. Bounds (5 parts) [25 points]
For each of the following problems, give the best possible bound. Since the upper and lower
asymptotic bounds are the same for each answer, you can just give the bound without surrounding
� ’s, �’s, or �’s. If you wish to justify your answer, please do so.
(a) The depth of a circuit for multiplying two �-bit numbers. ���� �� Solution: (b) The depth of a circuit to rerepresent the sum of � �-bit binary numbers as the sum of � �� � ��-bit binary numbers. Solution: ���� 6.896 Final Examination Name (c) The time to sort � numbers on an � � � mesh-of-trees network. ���� �� Solution: (d) The �-dimensional volume to permute any � wires. ������ � Solution: (e) The time to simulate an �-node 3-dimensional mesh on an ��� �� � ��-area area-universal
fat-tree with � leaves. (That is, simulate on a basic area-universal fat-tree, not the more
c...
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